Lagrange Method. Why do we want the axioms 1 – 7 of consumer theory? Answer: We like an easy life! By that we mean that we want well behaved demand curves.

Lagrange Method

Lagrange Method

• Why do we want the axioms 1 – 7 of consumer theory?

• Answer: We like an easy life!

By that we mean that we want well behaved demand curves.

Let’s look at a Utility Function: U = U(,y)

Take the total derivative:

dy

y

yxUdx

x

yxUdU

,,

1131.2 dU

0233.2 dU

For example if MUx = 2 MUy = 3

dyMUdxMUdU yx

Look at the special case of the total derivative along a given indifference curve:

10U

dy

dx

0

dyy

Udx

x

UdU

dyy

Udx

x

U

dx

dy

yUxU

xMRSydx

dy

MUy

MUx,

• Taking the total derivative of a B.C. yields

Px dx + Py dy = dM

• Along a given B.C. dM = 0

Px dx + Py dy = 0

MyPxP yx

Py

Px

dx

dy

y

x

=> Slope of the Indifference Curve

= Slope of the Budget Constraint

Py

Px

yUxU

MU

MU

y

x

Equilibrium

x

y

We have a general method for finding a point of tangency between an Indifference Curve and the

Budget Constraint:

The Lagrange Method

Widely used in Commerce, MBA’s

and Economics.

y

xy

x

u0

u1u2

Idea: Maximising U(x,y) is like climbing happiness mountain.

But we are restricted by how high we can go

since must stay on BC - (path on mountain).

y

x

u0

u1

u2

So to move up happiness Mountain is subject to being on a

budget constraint path.

Maximize U (x,y) subject to Pxx+ Pyy=M

yPxPMyxULMax yxyx

),(,

Known: Px, Py & M Unknowns: x,y,l

3 Equations: 3 Unknowns: Solve

32.. yxUge

yPxPMyxLMax yx 32

xx PxyL 32.1

PyyxLy 223.2

yPxPML yx .3

= 0

= 0

= 0

Trick:

But:

32xy

xx

yx 22 32

x

yx 322 Note:

U

Known: Px, Py & M Unknowns: x,y,

3 Equations: 3 Unknowns: Solve

32 yxU

yPxPMyxLMax yx 32

xx PxyL 32.1

PyyxLy 223.2

yPxPML yx .3

= 0

= 0

= 0

xPx

U 2

Pyy

U 3

32 yxU

yPxPMyxLMax yx 32

02

.1 xx Px

UL

03

.2 Pyy

ULy

0.3 yPxPML yx

xPx

U 2

Pyy

U 3

yx yP

U

xP

U 32

yx yPxP

32

y

x

P

P

x

y

3

2

32 yxU

yPxPMyxLMax yx 32

02

.1 xx Px

UL

03

.2 Pyy

ULy

xPx

U 2

Pyy

U 3

Notice:

U = x2 y3

xMUx

U

x

yxxy

x

U

22

232

3yMU

y

U

y

U

3

MUy

MUx

yUxU

yUxU

3

2

U

y

x

U

3

2

x

y

3

2

Recall Slope of Budget Constraint = y

x

P

P

y

x

y

x

P

P

x

y

MU

MUSo

3

2

<=> Slope of the Indifference Curve

Slope of IC = slope of BC

Back to the Problem:

Py

Px

x

y

3

2

yPxPM yx

xP

Py

y

x2

3

xP

PPxPM

y

xyx

2

3

y

x

P

P

x

y

3

2

+

But

But +

Back to the Problem:

xP

Py

y

x2

3

xPxPM xx 2

3

y

x

P

P

x

y

3

2

xPM x

2

31

+

But

But +

Py

Px

x

y

3

2

yPxPM yx

xPM x

2

5

xP

Mx

5

2

xx

D PMxP

Mx ,

5

2

So the Demand Curve for x when U=x2y3

If M=100:

x

D

Px

5

200

Demand Curve for X

0

2

4

6

8

10

12

Quantity

Pri

ce

of

x

Price of x 10 8 5 2

4 5 8 20

PxxD

10 4

8 5

5 8

2 20

Recall that: U = x2 y3

Let: U = xa yb

For Cobb - Douglas Utility Function

xPM

x

D

P

M

ba

ax

32

2

x

D

P

Mx

5

2

Note that: Cobb-Douglas is a special result

In general: MPPxx yx ,,

MPPyy yx ,,

MPxx x ,

MPyy y ,

For Cobb - Douglas:

Why does the demand for x not depend on py?

Share of x in income =

M

Pxs x

x

.

In this example: M

Ps x

x M

P

P

Ms x

xx

5

2

5

2

5

2

M

P

P

Ms x

xx Constant

Similarly share of y in income is constant:

5

3

5

3

M

P

P

Ms y

yy

So if the share of x and y in income is constant => change in Px only effects demand for x in C.D.

yPxPMyxL yx 32

2U

1U

M

U

ML So l tells us the change in U as M rises

Increase M

Increase from U1 to U2

in constraint

ConstraintObjective fn

in objective fn